Limit theorems for nonlinear functionals of Volterra processes via white noise analysis

DOI:10.3150/10-BEJ258

Limit theorems for nonlinear functionals of Volterra processes via white noise analysis

S É BA S T I E N DA R S E S¹, I VA N N O U R D I N²and DAV I D N UA L A RT³

1Université Aix-Marseille I, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France.

E-mail:darses@cmi.univ-mrs.fr

2Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie (Paris VI), Boîte Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France. E-mail:ivan.nourdin@upmc.fr

3Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA.

E-mail:nualart@math.ku.edu

By means of white noise analysis, we prove some limit theorems for nonlinear functionals of a given Volterra process. In particular, our results apply to fractional Brownian motion (fBm) and should be com- pared with the classical convergence results of the 1980s due to Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu, as well as the recent advances concerning the construction of a Lévy area for fBm due to Coutin, Qian and Unterberger.

Keywords:fractional Brownian motion; limit theorems; Volterra processes; white noise analysis

1. Introduction

FixT >0 and letB=(B_t)_t≥0be a fractional Brownian motion with Hurst indexH ∈(0,1), defined on some probability space(,B, P ). Assume thatBis the completedσ-field generated byB. Fix an integerk≥2 and, forε >0, consider

Gε=ε^{−k(1−H )} _T

0

hk

B_u+ε−B_u ε^H

du. (1.1)

Here, and in the rest of this paper,

h_k(x)=(−1)^ke^x2/2 d^k

dx^k(e^−x2/2) (1.2)

stands for thekth Hermite polynomial. We haveh2(x)=x²−1,h3(x)=x³−3xand so on.

Since the seminal works [3,6,7,19,20] by Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu, the following three convergence results are classical:

• ifH <1−_2k¹, then

(B_t)_t∈[0,T], ε^{k(1−H )−1/2}G_εLaw

−→ε→0

(B_t)_t∈[0,T], N

, (1.3)

whereN∼N(0, T ×k!_T

0 ρ^k(x)dx)is independent ofB, withρ(x)=₂¹(|x+1|^2H+ |x− 1|^2H−2|x|^2H);

1350-7265 © 2010 ISI/BS

• ifH=1−_2k¹, then

(B_t)_t∈[0,T], G_ε log(1/ε)

Law

−→ε→0

(B_t)_t∈[0,T], N

, (1.4)

whereN∼N(0, T ×2k!(1−_2k¹)^k(1−¹_k)^k)is independent ofB;

• ifH >1−_2k¹, then

G_ε^L

2()

−→ε→0Z^(k)_T , (1.5)

whereZ_T^(k)denotes the Hermite random variable of orderk; see Section4.1for its definition.

Combining (1.3) with the fact that sup_0<ε≤1E[|ε^{k(1−H )−1/2}Gε|^p]<∞for allp≥1 (use the boundedness of Var(ε^{k(1−H )−1/2}Gε)and a classical hypercontractivity argument), we have, for allη∈L²()and ifH <1−_2k¹, that

ε^{k(1−H )−1/2}E[ηG_ε] −→

ε→0E(ηN )=E(η)E(N )=0

(a similar statement holds in the critical case H =1− _2k¹). This means that ε^{k(1−H )−1/2}G_ε convergesweaklyinL²()to zero. The following question then arises. Is there a normalization ofG_εensuring that it converges weaklytowards anon-zerolimit whenH≤1−_2k¹? If so, then what can be said about the limit? The first purpose of the present paper is to provide an answer to this question in the framework ofwhite noise analysis.

In [14], it is shown that for allH ∈(0,1), the time derivativeB˙ (called thefractional white noise) is a distribution in the sense of Hida. We also refer to Bender [1], Biaginiet al.[2] and references therein for further works on the fractional white noise.

Since we haveE(Bu+ε−Bu)²=ε^2H, observe thatGεdefined in (1.1) can be rewritten as Gε=

T 0

B_u+ε−B_u ε

k

du, (1.6)

where(. . .)^k denotes thekth Wick product. In Proposition9 below, we will show that for all H∈(¹₂−¹_k,1),

εlim→0

_T

0

B_u+ε−B_u ε

k

du= _T

0

B˙_u^kdu, (1.7)

where the limit is in the(S)^∗sense.

In particular, we observe two different types of asymptotic results forGε when H ∈(¹₂ −

1

k,1−_2k¹): convergence (1.7) in(S)^∗to a Hida distribution, and convergence (1.3) in law to a normal law, with rateε^{1/2−k(1−H )}. On the other hand, whenH∈(1−_2k¹,1), we obtain from (1.5) that the Hida distribution_T

0 B˙_s^kdsturns out to be the square-integrable random variable Z_T^(k), which is an interesting result in its own right.

In Proposition 4, the convergence (1.7) in (S)^∗ is proved for a general class of Volterra processes of the form

t 0

K(t, s)dWs, t≥0, (1.8)

whereW stands for a standard Brownian motion, provided the kernelKsatisfies some suitable conditions; see Section3.

We also provide a new proof of the convergence (1.3) based on the recent general criterion for the convergence in distribution to a normal law of a sequence of multiple stochastic integrals established by Nualart and Peccati [15] and by Peccati and Tudor [17], which avoids the classical method of moments.

In two recent papers [9,10], Marcus and Rosen have obtained central and non-central limit theorems for a functional of the form (1.1), where B is a mean zero Gaussian process with stationary increments such that the covariance function ofB, defined byσ²(|t−s|)=Var(Bt− Bs), is either convex (plus some additional regularity conditions), concave or given byσ²(h)= h^r with 1< r <2. These theorems include the convergence (1.3) and, unlike our simple proof, are based on the method of moments.

In the second part of the paper, we develop a similar analysis for functionals of two indepen- dent fractional Brownian motions (or, more generally, Volterra processes) related to the Lévy area. More precisely, consider twoindependentfractional Brownian motionsB⁽¹⁾andB⁽²⁾with (for simplicity) the same Hurst indexH∈(0,1). We are interested in the convergence, asε→0, of

G_ε:=

T 0

B_u⁽¹⁾B_u⁽²⁾_+ε−B_u⁽²⁾

ε du (1.9)

and

G˘ε:=

_T

0

u 0

B_v⁽¹⁾_+ε−Bv⁽¹⁾

ε dv

B_u⁽²⁾_+ε−B_u⁽²⁾

ε du. (1.10)

Note that G_ε coincides with the ε-integral associated with the forward Russo–Vallois integral T

0 B⁽¹⁾d⁻B⁽²⁾; see, for example, [18] and references therein. Over the last decade, the conver- gences ofG_ε andG˘_ε (or of related families of random variables) have been intensively studied.

Sinceε⁻¹_u

0(B_v⁽¹⁾_+ε−B_v⁽¹⁾)dv converges pointwise toB_u⁽¹⁾ for anyu, we could think that the asymptotic behaviors ofG_ε andG˘_ε are very close asε→0. Surprisingly, this is not the case.

Actually, only the result forG˘εagrees with the seminal result of Coutin and Qian [4] (that is, we have convergence ofG˘_ε inL²()if and only ifH >1/4) and with the recent result by Unter- berger [21] (that is, adequately renormalized,G˘εconverges in law ifH <1/4). More precisely:

• ifH <1/4, then

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], ε^1/2−2HG˘_εLaw

−→ε→0

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], N

, (1.11)

whereN∼N(0, Tσ˘_H²)is independent of(B⁽¹⁾, B⁽²⁾)and

˘

σ_H² = 1

4(2H+1)(2H+2)

R(|x+1|^2H+ |x−1|^2H−2|x|^2H)

×(2|x|^2H+2− |x+1|^2H+2− |x−1|^2H+2)dx;

• ifH=1/4, then

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], G˘_ε log(1/ε)

Law

−→ε→0

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], N

, (1.12)

whereN∼N(0, T /8)is independent of(B⁽¹⁾, B⁽²⁾);

• ifH >1/4, then

G˘_ε^L

2()

−→ε→0

T 0

B_u⁽¹⁾ ˙B_u⁽²⁾du= T

0

B_u⁽¹⁾dB_u⁽²⁾; (1.13)

• for allH∈(0,1), we have

G˘_ε−→^(S)∗

ε→0

T 0

B_u⁽¹⁾ ˙B_u⁽²⁾du. (1.14)

However, forG_ε, we have, in contrast:

• ifH <1/2, then

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], ε^1/2−HG_εLaw

−→ε→0

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], N×S

, (1.15)

where

S= _∞

0

(|x+1|^2H+ |x−1|^2H−2|x|^2H)dx× T

0

B_u⁽¹⁾2

du andN∼N(0,1), independent of(B⁽¹⁾, B⁽²⁾);

• ifH≥1/2, then

G_ε^L

2()

−→ε→0

_T

0

B_u⁽¹⁾ ˙B_u⁽²⁾du= _T

0

B_u⁽¹⁾dB_u⁽²⁾; (1.16)

• for allH∈(0,1), we have

G_ε−→^(S)∗

ε→0

_T

0

B_u⁽¹⁾ ˙B_u⁽²⁾du. (1.17)

Finally, we study the convergence, asε→0, of the so-called ε-covariation(following the terminology of [18]) defined by

G_ε:=

_T

0

B_u⁽¹⁾_+ε−B_u⁽¹⁾

ε ×B_u⁽²⁾_+ε−B_u⁽²⁾

ε du (1.18)

and we get:

• ifH <3/4, then

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], ε^3/2−2HGε

Law

−→ε→0

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], N

(1.19) withN∼N(0, Tσ_H²)independent of(B⁽¹⁾, B⁽²⁾)and

σ_H² =1

4

R(|x+1|^2H+ |x−1|^2H−2|x|^2H)²dx;

• ifH=3/4, then

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], Gε

log(1/ε) Law

−→ε→0

B_t⁽¹⁾, B_t⁽²⁾

t∈[0,T], N

(1.20) withN∼N(0,9T /32)independent of(B⁽¹⁾, B⁽²⁾);

• ifH >3/4, then

G_ε^L

2()

−→ε→0

_T

0

B˙_u⁽¹⁾ ˙B_u⁽²⁾du; (1.21)

• for allH∈(0,1), we have

G_ε−→^(S)∗

ε→0

T 0

B˙_u⁽¹⁾ ˙B_u⁽²⁾du. (1.22) The paper is organized as follows. In Section2, we introduce some preliminaries on white noise analysis. Section3 is devoted to the study, using the language and tools of the previous section, of the asymptotic behaviors ofG_ε,G_εandG_εin the (more general) context whereBis a Volterra process. Section4is concerned with the fractional Brownian motion case. In Section5 (resp., Section6), we prove (1.3) and (1.4) (resp., (1.11), (1.12), (1.15), (1.19) and (1.20)).

2. White noise functionals

In this section, we present some preliminaries on white noise analysis. The classical approach to white noise distribution theory is to endow the space of tempered distributionsS(R)with a Gaussian measurePsuch that, for any rapidly decreasing functionη∈S(R),

S(R)

e^ix,ηP(dx)=e^−|η|20/2.

Here,| · |0denotes the norm inL²(R)and·,·the dual pairing betweenS(R)andS(R). The existence of such a measure is ensured by Minlos’ theorem [8].

In this way, we can consider the probability space (,B,P), where=S(R). The pair- ing x, ξ can be extended, using the norm of L²(), to any function ξ ∈ L²(R). Then,

W_t= ·,1_[0,t]is a two-sided Brownian motion (with the convention that1_[0,t]= −1_[t,0]ift <0) and for anyξ∈L²(R),

·, ξ = _∞

−∞ξdW=I₁(ξ ) is the Wiener integral ofξ.

Let∈L²(). The classical Wiener chaos expansion ofsays that there exists a sequence of symmetric square-integrable functionsφ_n∈L²(Rⁿ)such that

= ∞ n=0

In(φn), (2.1)

whereI_ndenotes the multiple stochastic integral.

2.1. The space of Hida distributions

Let us recall some basic facts concerning tempered distributions. Let(ξ_n)^∞_n=0be the orthonormal basis ofL²(R)formed by the Hermite functions given by

ξ_n(x)=π^−1/4(2ⁿn!)^−1/2e^−x2/2h_n(x), x∈R, (2.2) whereh_nare the Hermite polynomials defined in (1.2). The following two facts can immediately be checked: (a) there exists a constantK₁>0 such that ξ_n∞≤K₁(n+1)^−1/12; (b) since ξ_n=

n

2ξ_n−1−

n+1

2 ξ_n+1, there exists a constantK₂>0 such thatξ_n∞≤K₂n^5/12.

Consider the positive self-adjoint operatorA (whose inverse is Hilbert–Schmidt) given by A= −_dx^d22 +(1+x²). We haveAξ_n=(2n+2)ξn.

For anyp≥0, define the spaceSp(R)to be the domain of the closure ofA^p. Endowed with the norm |ξ|p:= |A^pξ|0, it is a Hilbert space. Note that the norm| · |p can be expressed as follows, if one uses the orthonormal basis(ξ_n):

|ξ|²p= ∞ n=0

ξ, ξn²(2n+2)^2p.

We denote byS_p(R)the dual ofSp(R). The norm inS_p(R)is given by (see [16], Lemma 1.2.8)

|ξ|²₋p= ∞ n=0

|ξ, A^−pξn|²= ∞ n=0

ξ, ξn²(2n+2)^−2p

for anyξ∈S_p(R). One can show that the projective limit of the spacesSp(R),p≥0, isS(R), that the inductive limit of the spacesSp(R),p≥0, isS(R)and that

S(R)⊂L²(R)⊂S(R)

is a Gel’fand triple.

We can now introduce the Gel’fand triple

(S)⊂L²()⊂(S)^∗,

via the second quantization operator(A). This is an unbounded and densely defined operator onL²()given by

(A)=^∞

n=0

I_n(A^⊗nφ_n),

wherehas the Wiener chaos expansion (2.1). Ifp≥0, then we denote by(S)_p the space of random variables∈L²()with Wiener chaos expansion (2.1) such that

^pp:=E[|(A)^p|²] = ∞ n=0

n!|φn|²p<∞.

In the above formula,|φ_n|p denotes the norm inSp(R)^⊗n. The projective limit of the spaces (S)_p,p≥0, is called the space of test functions and is denoted by(S). The inductive limit of the spaces(S)_−p,p≥0, is called the space of Hida distributions and is denoted by(S)^∗. The elements of(S)^∗ are calledHida distributions. The main example is the time derivative of the Brownian motion, defined asW˙_t= ·, δ_t. One can show that|δ_t|₋p<∞for somep >0.

We denote by, the dual pairing associated with the spaces(S)and(S)^∗. On the other hand (see [16], Theorem 3.1.6), for any∈(S)^∗, there existφ_n∈S(Rⁿ)such that

, =^∞

n=0

n!φ_n, ψ_n,

where=_∞

n=0In(ψn)∈(S). Moreover, there existsp >0 such that ²_−p=^∞

n=0

n!|φ_n|²_−p.

Then, with a convenient abuse of notation, we say thathas a generalized Wiener chaos expan- sion of the form (2.1).

2.2. The S -transform

A useful tool to characterize elements in(S)^∗ is the S-transform. The Wick exponential of a Wiener integralI₁(η),η∈L²(R), is defined by

:e^I1(η): =e^{I1(η)−|η|20/2}.

TheS-transform of an element∈(S)^∗is then defined by S()(ξ )=

,:e^{I1(ξ )}: ,

whereξ∈S(R). One can easily see that theS-transform is injective on(S)^∗.

If∈L²(), thenS()(ξ )=E[:e^{I1(ξ )}:]. For instance, theS-transform of the Wick ex- ponential is

S

:e^I1(η):

(ξ )=e^η,ξ. Also,S(Wt)(ξ )=t

0ξ(s)dsandS(W˙t)(ξ )=ξ(t ).

Suppose that∈(S)^∗has a generalized Wiener chaos expansion of the form (2.1). Then, for anyξ∈S(R),

S()(ξ )=^∞

n=0

φ_n, ξ^⊗n, where the series converges absolutely (see [16], Lemma 3.3.5).

The Wick product of two functionals=_∞

n=0In(ψn)and=_∞

n=0In(φn)belonging to (S)^∗is defined as

=

∞ n,m=0

I_n+m(ψ_n⊗φ_m).

It can be proven that∈(S)^∗. The following is an important property of theS-transform:

S()(ξ )=S()(ξ )S()(ξ ). (2.3)

If,andbelong toL²(), then we haveE[] =E[]E[].

The following is a useful characterization theorem.

Theorem 1. A functionF is theS-transform of an element∈(S)^∗if and only if the following conditions are satisfied:

(1) for anyξ, η∈S,z→F (zξ+η)is holomorphic onC;

(2) there exist non-negative numbersK, aandpsuch that for allξ∈S,

|F (ξ )| ≤Kexp(a|ξ|²p).

Proof. See [8], Theorems 8.2 and 8.10.

In order to study the convergence of a sequence in(S)^∗, we can use itsS-transform, by virtue of the following theorem.

Theorem 2. Let _n ∈(S)^∗ and S_n=S(_n).Then, _n converges in (S)^∗ if and only if the following conditions are satisfied:

(1) limn→∞S_n(ξ )exists for eachξ∈S;

(2) there exist non-negative numbersK, aandpsuch that for alln∈N,ξ∈(S),

|S_n(ξ )| ≤Kexp(a|ξ|²_p).

Proof. See [8], Theorem 8.6.

3. Limit theorems for Volterra processes

3.1. One-dimensional case

Consider a Volterra processB=(B_t)_t≥0of the form B_t=

_t

0

K(t, s)dWs, (3.1)

whereK(t, s)satisfies_t

0K(t, s)²ds <∞for allt >0 andW is the Brownian motion defined on the white noise probability space introduced in the last section. Note that theS-transform of the random variableB_t is given by

S(Bt)(ξ )= _t

0

K(t, s)ξ(s)ds (3.2)

for anyξ ∈S(R). We introduce the following assumptions on the kernelK:

(H1) Kis continuously differentiable on{0< s < t <∞}and for anyt >0, we have _t

0

∂K

∂t (t, s)

(t−s)ds <∞;

(H2) k(t )=t

0K(t, s)dsis continuously differentiable on(0,∞).

Consider the operatorK₊defined by K₊ξ(t )=k(t)ξ(t)+

_t

0

∂K

∂t (t, r)

ξ(r)−ξ(t ) dr,

wheret >0 andξ∈S(R). From Theorem1, it follows that the linear mappingξ→K₊ξ(t )is theS-transform of a Hida distribution. More precisely, according to [14], define the function

C(t )= |k(t)| + _t

0

∂K

∂t (t, r)

(t−r)dr, t≥0, (3.3)

and observe that the following estimates hold (recall the definition (2.2) ofξ_n):

|K₊ξ(t )| ≤C(t )(ξ_∞+ ξ_∞)

≤C(t ) ∞ n=0

|ξ, ξ_n|(ξ_n∞+ ξ_n∞)

≤C(t )M ∞ n=0

|ξ, ξ_n|(n+1)^5/12 (3.4)

≤C(t )M ^∞

n=0

|ξ, ξ_n|²(2n+2)^{17/6 ∞}

n=0

(n+1)⁻²

=C(t )M|ξ|17/12

for some constantsM >0 whose values are not always the same from one line to the next.

We have the following preliminary result.

Lemma 3. Fix an integerk≥1.LetB be a Volterra process with kernelK satisfying the con- ditions(H1)and (H2).Assume,moreover,that C defined by(3.3)belongs to L^k([0, T]).The functionξ →_T

0 (K₊ξ(s))^kds is then the S-transform of an element of(S)^∗.This element is denoted by_T

0 B˙_u^kdu.

Proof. We use Theorem1. Condition (1) therein is immediately checked, while for condition (2), we just write, using (3.4),

_T

0

(K₊ξ(s))^kds ≤

_T

0

|K₊ξ(s)|^kds≤M|ξ|17/12

_T

0

C^k(s)ds.

Fix an integerk≥1 and consider the following, additional, condition.

(H^k₃) The maximal functionD(t)=sup_0<ε≤ε0 ¹_εt+ε

t C(s)ds belongs toL^k([0, T])for any T >0 and for someε0>0.

We can now state the main result of this section.

Proposition 4. Fix an integerk≥1.LetB be a Volterra process with kernelK satisfying the conditions(H1),(H2)and(H^k₃).The following convergence then holds:

T 0

B_u+ε−B_u ε

k

du−→^(S)∗

ε→0

T 0

B˙_u^kdu.

Proof. Fixξ∈S(R)and set S_ε(ξ )=S

T 0

B_u+ε−B_u ε

k

du

(ξ ).

From linearity and property (2.3) of theS-transform, we obtain S_ε(ξ )=

_T

0

(S(B_u+ε−B_u)(ξ ))^k

ε^k du. (3.5)

Equation (3.2) yields

S(B_u+ε−B_u)(ξ )= _u+ε

0

K(u+ε, r)ξ(r)dr− _u

0

K(u, r)ξ(r)dr. (3.6) We claim that

_u+ε

0

K(u+ε, r)ξ(r)dr− _u

0

K(u, r)ξ(r)dr= _u+ε

u

K₊ξ(s)ds. (3.7) Indeed, we can write

_u+ε

u

K₊ξ(s)ds= _u+ε

u

k(s)ξ(s)ds+ _u+ε

u

_s

0

∂K

∂s (s, r)

ξ(r)−ξ(s) dr

ds

(3.8)

=A⁽¹⁾_u +A⁽²⁾_u . We have, using Fubini’s theorem, that

A⁽²⁾_u = − _u+ε

u

ds _s

0

dr∂K

∂s(s, r) _s

r

dθ ξ(θ )

(3.9)

= − _u+ε

0

dθ ξ(θ ) _θ

0

dr

K(u+ε, r)−K(θ∨u, r) .

This can be rewritten as A⁽²⁾_u = −

_u

0

K(u+ε, r)−K(u, r)

ξ(u)−ξ(r) dr

(3.10)

− _u+ε

u

dθ ξ(θ ) _θ

0

dr

K(u+ε, r)−K(θ, r) .

On the other hand, integration by parts yields A⁽¹⁾_u =ξ(u+ε)

_u+ε

0

K(u+ε, r)dr

(3.11)

−ξ(u) _u

0

K(u, r)dr− _u+ε

u

ds ξ(s) _s

0

dr K(s, r).

Therefore, adding (3.11) and (3.10) yields A⁽¹⁾_u +A⁽²⁾_u =ξ(u+ε)

_u+ε

0

K(u+ε, r)dr−ξ(u) _u

0

K(u, r)dr

− _u

0

K(u+ε, r)−K(u, r)

ξ(u)−ξ(r)

dr (3.12)

− _u+ε

u

dθ ξ(θ ) _θ

0

K(u+ε, r)dr.

Note that, by integrating by parts, we have

− _u+ε

u

dθ ξ(θ ) _θ

0

K(u+ε, r)dr

= −ξ(u+ε) _u+ε

0

K(u+ε, r)dr+ξ(u) _u

0

K(u+ε, r)dr (3.13) +

_u+ε

u

K(u+ε, r)ξ(r)dr.

Thus, substituting (3.13) into (3.12), we obtain A⁽¹⁾_u +A⁽²⁾_u =

_u+ε

0

K(u+ε, r)ξ(r)dr− _u

0

K(u, r)ξ(r)dr, which completes the proof of (3.7). As a consequence, from (3.5)–(3.7), we obtain

S_ε(ξ )= T

0

1 ε

u+ε u

K₊ξ(s)ds k

du.

On the other hand, using (3.4) and the definition of the maximal functionD, we get sup

0<ε≤ε₀

1 ε

_u+ε

u

K₊ξ(s)ds

^k≤M^k|ξ|^k17/12 sup

0<ε≤ε₀

1 ε

_u+ε

u

C(s)ds k

(3.14)

=M^k|ξ|^k_17/12D^k(u).

Therefore, using hypothesis(H^k₃)and the dominated convergence theorem, we have

εlim→0S_ε(ξ )= T

0

(K₊ξ(s))^kds. (3.15)

Moreover, since|S_ε(ξ )| ≤M^k|ξ|^k_17/12T

0 D^k(u)dufor all 0< ε≤ε₀(see (3.14)), conditions (1) and (2) in Proposition4are fulfilled. Consequently,ε^−k_T

0 (B_u+ε−B_u)^kduconverges in(S^∗) asε→0.

To complete the proof, it suffices to observe that the right-hand side of (3.15) is, by definition (see Lemma3), theS-transform of_T

0 B˙_s^kds.

In [14], it is proved that under some additional hypotheses, the mappingt→B_t is differen- tiable from(0,∞)to(S)^∗and that its derivative, denoted by B˙_t, is a Hida distribution whose S-transform isK₊ξ(t ).

3.2. Bidimensional case

LetW=(W_t)_t∈Rbe a two-sided Brownian motion defined in the white noise probability space (S(R),B,P). We can consider two independent standard Brownian motions as follows: for t≥0, we setW_t⁽¹⁾=W_t andW_t⁽²⁾=W_−t.

In this section, we consider a bidimensional processB=(B_t⁽¹⁾, B_t⁽²⁾)_t≥0, whereB⁽¹⁾andB⁽²⁾ are independent Volterra processes of the form

B_t⁽ⁱ⁾= _t

0

K(t, s)dW_s⁽ⁱ⁾, t≥0, i=1,2. (3.16) For simplicity only, we work with the same kernelKfor the two components.

First, using exactly the same lines of reasoning as in the proof of Lemma3, we get the follow- ing result.

Lemma 5. Let B=(B_t⁽¹⁾, B_t⁽²⁾)_t≥0 be given as above,with a kernel K satisfying the condi- tions(H1)and (H2).Assume,moreover,that C defined by(3.3)belongs toL²([0, T])for any T >0.We then have the following results:

(1) the function ξ →_T

0 (_u

0 K₊ξ(−y)dy)K₊ξ(u)du is the S-transform of an element of (S)^∗,denoted by_T

0 B_u⁽¹⁾ ˙B_u⁽²⁾du;

(2) the functionξ→_T

0 K₊ξ(−u)K₊ξ(u)duis theS-transform of an element of(S)^∗,de- noted by_T

0 B˙_u⁽¹⁾ ˙B_u⁽²⁾du.

We can now state the following result.

Proposition 6. LetB=(B_t⁽¹⁾, B_t⁽²⁾)_t≥0be given as above,with a kernelKsatisfying the condi- tions(H1),(H2)and(H²₃).The following convergences then hold:

_T

0

B_u⁽¹⁾B_u⁽²⁾_+ε−B_u⁽²⁾ ε du−→^(S)∗

ε→0

_T

0

B_u⁽¹⁾ ˙B_u⁽²⁾du, T

0

u 0

B_v⁽¹⁾_+ε−Bv⁽¹⁾

ε dv

B_u⁽²⁾_+ε−B_u⁽²⁾ ε du−→^(S)∗

ε→0

T 0

B_u⁽¹⁾ ˙B_u⁽²⁾du, _T

0

B_u⁽¹⁾_+ε−B_u⁽¹⁾

ε ×B_u⁽²⁾_+ε−B_u⁽²⁾ ε du−→^(S)∗

ε→0

_T

0

B˙_u⁽¹⁾ ˙B_u⁽²⁾du.

Proof. Set

G_ε= T

0

B_u⁽¹⁾B_u⁽²⁾_+ε−B_u⁽²⁾

ε du=

T 0

B_u⁽¹⁾B_u⁽²⁾_+ε−B_u⁽²⁾

ε du.

From linearity and property (2.3) of theS-transform, we have S( G_ε)(ξ )=1

ε _T

0

S B_u⁽¹⁾

(ξ )S

B_u⁽²⁾_+ε−B_u⁽²⁾ (ξ )du so that

S(Gε)(ξ )= _T

0

u 0

K₊ξ(−y)dy 1

ε _u+ε

u

K₊ξ(x)dx

du.

Therefore, using (3.4) and (3.14), we can write

|S(G_ε)(ξ )| ≤M²|ξ|²17/12

_T

0

u 0

C(t )dt

D(u)du

≤M²|ξ|²17/12

T 0

u 0

D(t)dt

D(u)du

=1

2M²|ξ|²17/12

T 0

D(u)du 2

≤ T

2M²|ξ|²17/12

_T

0

D²(u)du.

Hence, by the dominated convergence theorem, we get

εlim→0

S( Gε)(ξ )= T

0

u 0

K₊ξ(−y)dy

K₊ξ(u)du. (3.17)

The right-hand side of (3.17) is theS-transform ofT

0 Bu⁽¹⁾ ˙Bu⁽²⁾du, due to Lemma5. Therefore, by Theorem2, we obtain the desired result in point (1).

The proofs of the other two convergences follow exactly the same lines of reasoning and are

therefore left to the reader.

4. Fractional Brownian motion case

4.1. One-dimensional case

Consider a (one-dimensional) fractional Brownian motion (fBm)B=(B_t)_t≥0 of Hurst index H∈(0,1). This means thatBis a zero mean Gaussian process with covariance function

R_H(t, s)=E(B_tB_s)=¹₂(t^2H+s^2H− |t−s|^2H).